Working with Probabilities Physics 115a (Slideshow 1) A. Albrecht

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Working with Probabilities Physics 115a (Slideshow 1) A. Albrecht

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Working with Probabilities Physics 115a (Slideshow 1) A. Albrecht

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Working with Probabilities

Physics 115a (Slideshow 1)

A. Albrecht

These slides related to Griffiths section 1.3

Consider the following group of people in a room:

Histogram Form

Consider the following group of people in a room:

Total people = 14

Consider the following group of people in a room:

Total people = 14

Consider the following group of people in a room:

Total people = 14

Consider the following group of people in a room:

Total people = 14

Consider the following group of people in a room:

Total people = 14

Consider the following group of people in a room:

Total people = 14

Consider the following group of people in a room:

Total people = 14

Probability Histogram

Number Histogram

NB: The probabilities for ages not listed are all zero

Total people = 14

Assuming Age<20, what is the probability of finding each age?

Total people = 14

Assuming Age<20, what is the probability of finding each age?

Total people = 14

Assuming Age<20, what is the probability of finding each age?

Total people = 14

Assuming no age constraint, what is the probability of finding each age?

Related to collapse of the waveunction (“changing the question”)

Total people = 14

Assuming Age<20, what is the probability of finding each age?

Related to collapse of the waveunction (“changing the question”)

Total people = 14

Consider a different room with different people:

Total people = 15

Consider a different room with different people:

Total people = 15

Red Room Numbers

Red Room Probabilities

Combine Red and Blue rooms

Total people = 29

- Lessons so far
- A simple application of probabilities
- Normalization
- “Re-Normalization” to answer a different question
- Adding two “systems”.
- All of the above are straightforward applications of intuition.

Expectation Values

Most probable answer = 25

Median = 23

Average = 21

Most probable answer = 25

Median = 23

Average = 21

Lesson: Lots of different types of questions (some quite similar) with different answers. Details depend on the full probability distribution.

Average (mean):

- Standard QM notation
- Called “expectation value”
- NB in general (including the above) the “expectation value” need not even be possible outcome.

Average (number squared)

Careful: In general

In general, the average (or expectation value) of some function f(j) is

The “width” of a probability distribution

Discuss eqns 1.10 through 1.13 at board

Continuous Variables

Continuous Variables

Why not measure age in weeks?

Blue room in weeks

Blue room in weeks

Conclusion: Blue room in weeks not very useful/intuitive

Another case where a measure of age in weeks might by useful:

The ages of students taking health in the 8th grade in a large school district (3000 students).